**Quantum field theory **predicts that the temperature of empty space should depend on the observer’s motion, increasing proportionally with acceleration. Here I attempt an accessible introduction to this striking effect, related to **Hawking radiation **and discovered independently by **Fulling, Davies, **and** Unruh**, assuming only sophomore-level physics (including hyperbolic functions) with some assistance from **Mathematica**.

## Hyperbolic Motion

Constant acceleration in Newtonian mechanics is **parabolic**, while constant acceleration in Einsteinian mechanics is **hyperbolic** and asymptotic to light speed c = 1 in natural units. For 1+1-dimensional **Minkowski spacetime**, the *difference* in squared space and time displacements is the square of **proper time** displacement,

For *constant* **proper acceleration**, this has the solution

with velocity

v = \frac{dx}{dt} = \tanh a\tau \le 1,where for small times v \sim a \tau \sim a t. If t = 0 and a x = 1 at \tau = 0, then integration gives

a t = \sinh a \tau,\\a x = \cosh a\tau.Hyperbolic identities then imply

a^2 x^2 - a^2 t^2 = 1and

## Quantum Vacuum

Due to **Heisenberg indeterminacy**, electromagnetic fluctuations fill the vacuum. Consider a single such sinusoidal wave of angular frequency \omega_0 = k_0 in natural units. If you move at constant velocity, you observe the wave **doppler-shifted** to a different frequency. But if you move at constant acceleration, you observe the wave doppler-shifted to *a range* of frequencies corresponding to your range of velocities. For an accelerated observer at time \tau,

\phi[x,t] = \exp\left[i(k_0 x + \omega_0 t)\right] = \exp\left[i\omega_0(x + t)\right] = \exp\left[{i \frac{\omega_0}{a} e^{a \tau}} \right] = \phi[\tau].

Expand this waveform as a sum of **harmonics**

where the **Fourier components**

To** regularize** this divergent integral, subtract a tiny imaginary part i \epsilon from the angular frequency \omega to incorporate a decaying exponential factor e^{-\epsilon \tau} in the integrand, and zero it after integrating. Find

where \Gamma[n+1] = n! analytically continues the factorial function to the complex plane. The spectrum is the absolute square of the **Fourier transform**,

where the **Planck factor **suggests **Bose-Einstein statistics** and a thermal **photon** bath of **temperature** kT = a \hbar / 2\pi. In SI units,

where g_E is** Earth’s** **surface gravity**, and a **zeptokelvin** is very cool.

## Hawking-Unruh Temperature

Just prior to the 1970s work of Fulling, Davies, and Unruh, **Stephen Hawking** famously predicted that despite their reputations **black holes **should radiate with an effective temperature

where \kappa is the black hole’s surface gravity (observed at infinity). The Unruh and Hawking results may be linked by the **equivalence principle**, which equates acceleration and gravity, and by **event horizons**. The black hole horizon is a boundary that **causally** disconnects the interior from the exterior. Similarly, when you accelerate, a **Rindler horizon** appears a distance c^2/a \sim 1~\text{ly} \left(g_E / a \right) behind you, causally disconnecting you from a region of spacetime whose photons you can outrun (so long as your acceleration continues).